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first is a simple exercise in the analytic geometry of conic
sections and betrays no originality. The second, which fills only
six and a half pages in Hermite's collected works, is a horse of
quite a different colour. Its unassuming title is Considerations
on the algebraic solution of the equation of the fifth degree (trans-
llt is known', the modest mathematician of twenty begins,
'that Lagrange made the algebraic solution of the general equa-
tion of the fifth degree depend on the determination of a root
of a particular equation of the sixth degree, which he calls a
reduced equation [to-day, a "resolvent"]-----So that, if this
resolvent were decomposable into rational factors of the
second or third degrees, we should have the solution of the
equation of the fifth degree, I shall try to show that such a
decomposition is impossible.* Hermite not only succeeded in
his attempt - by a beautifully simple argument - but showed
also in doing so that he was an algebraist. VTith but a few slight
changes this short paper will do all that is required.
It may seem strange that a young man capable of genuine
mathematical reasoning of the calibre shown by Hermite in his
paper on the general quintic should find elementary mathe-
matics difficult. But it is not necessary to understand - or even
to have heard of - much of classical mathematics as it has
evolved in the course of its long history in order to be able to
follow or work creatively in the mathematics that has been
developed since 1800 and is still of living interest to mathe-
maticians. The geometrical treatment (synthetic) of conic sec-
tions of the Greeks, for instance, need not be mastered to-day
by anyone who wishes to follow modern geometry; nor need
any geometry at all be learned by one whose tastes are alge-
braic or arithmetical. To a lesser degree the same is true for
analysis, where such geometrical language as is used is of the
simplest and is neither necessary nor desirable if up-to-date
proofs are the object. As a last example, descriptive geometry,
of great use to designing engineers, is of practically no use
whatever to a working mathematician. Some quite difficult
subjects that are still mathematically alive require only a school
education in algebra and a clear head for their comprehensjou.